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Interest Rate Forwards and Futures

1Forward Rates The forward rate is the future zero rate implied by todays term structure of interest rates10/4/2009Bahattin Buyuksahin, Celso Brunetti22Implied Forward RateAn investor investing 2 years has a choice of buying a 2 year zero coupon bond paying (1+R0,2)2 or buying a 1 year bond paying (1+R0,1) for 1 year , and reinvesting the proceeds at the implied forward rate, R1,2 between years 1 and 2. The forward rate makes the investor indifferent between these alternatives. That is to say,

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3Implied Forward RateIn general, we have

This further implies, the implied forward zero-coupon bond price must be consistent with the implied forward interest rate.

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4Formula for Forward Rates (continuous case)Suppose that the zero rates for maturities T1 and T2 are R1 and R2 with both rates continuously compounded.The forward rate for the period between times T1 and T2 is

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5Calculation of Forward Rates 10/4/2009Bahattin Buyuksahin, Celso Brunetti6Zero Rate forForward Ratean n-year Investmentfor nth YearYear (n)(% per annum)(% per annum)110.0210.511.0310.811.4411.011.6511.111.56To illustrate this formula, consider the five year forward rate from the data in our table. T4= 11%, T5= 11.1%

This shows that if zero rate is upward sloping between T1 and T2 , so that R2 >R1, then RF>R2

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7Instantaneous Forward RateThe instantaneous forward rate for a maturity T is the forward rate that applies for a very short time period starting at T. It is

where R is the T-year rate. If we define P(0,T) as the price of bond maturing at time T and paying $1, so that 10/4/2009Bahattin Buyuksahin, Celso Brunetti8

8Term Structure of Interest RatesThe term structure of interest rates (or yield curve) is the relationship of the yield to maturity against bond term (maturity).

Typical shapes are: increasing (normal), decreasing, humped and flat.10/4/2009Bahattin Buyuksahin, Celso Brunetti9YieldMaturity9Upward vs Downward SlopingYield Curve For an upward sloping yield curve:Fwd Rate > Zero Rate > Par Yield

For a downward sloping yield curvePar Yield > Zero Rate > Fwd Rate10/4/2009Bahattin Buyuksahin, Celso Brunetti1010Theories of the Term StructureA number of theory have been proposed: Expectation Hypothesis, Liquidity Preference Theory, Preferred Habitats Theory, Segmentation Hypothesis.

Fabozzi (1998): Pure Expectation Hypothesis, Liquidity Preference Theory, Preferred Habitats Theory are different forms of the expectation theory ==> two major theories: expectation theory and market segmentation theory.10/4/2009Bahattin Buyuksahin, Celso Brunetti1111Theories of the Term Structure of Interest Rates (1)The Pure Expectation Hypothesis: Implied forward rates are unbiased expectations of future spot rates ==> a rising term structure indicate that market expects short-term rates to rise in the future; a flat term structure reflects expectations that the future short term structure will be constant; and so on; Hicks (1937). Problems: It neglects the risks inherent in investing in bonds: if forward rates were perfect predictors of future interest rates then the future prices of bonds will be known with certainty.The Liquidity Preference Theory (Keynes): Given that there is uncertainty, long bonds should have higher returns than short bonds ==> we should expect a risk premium arising out from investors liquidity preferences. It is consistent with the empirical results that yield curves are upward sloping ==> positive risk premium.

10/4/2009Bahattin Buyuksahin, Celso Brunetti1212Theories of the Term Structure of Interest Rates (2)The Preferred Habitat Theory: It adopts the view that the term structure is composed by two components: Expectations plus risk premium (= liquidity preference theory). However, the risk premium might be negative as well as positive to induce market participants to shift out of their preferred habitat (Modigliani & Sutch (1966)).

The Segmentation Hypothesis (Culbertson (1957)): It also recognises that investors have preferred habitat (= preferred habitat theory) ==> individuals have strong maturity preferences ==> there need be no relationship between bonds with different maturities ==> bonds with different maturities are traded in different markets.10/4/2009Bahattin Buyuksahin, Celso Brunetti1313

Forward Rate ArrangementsWe now consider the problem of a borrower who wishes to hedge against increases in the cost of borrowing. Consider a firm expecting to borrow $100m for 91 days, beginning 120 days from today, in December. The loan will be repaid in March.Suppose that the effective quarterly interest rate can be either be 1.5% or 2%, and the implied September 91-day forward rate is 1.8%.Depending on the interest rate, there is a variation of $0.5m in the borrowing cost. How can we hedge this uncertainty?10/4/2009Bahattin Buyuksahin, Celso Brunetti1414Forward Rate AgreementA forward rate agreement (FRA) is an agreement that a certain rate will apply to a certain principal during a certain future time period

A FRA is equivalent to an agreement where interest at a predetermined rate, RK is exchanged for interest at the market rate

An FRA can be valued by assuming that the forward interest rate is certain to be realizedFRAs are forward contract based on the interest rate, and as such do not entail the actual lending of money. Rather, the borrower who enters an FRA is paid if a reference rate is above the FRA rate, and the borrower pays if the reference rate is below the FRA rate. The actual borrowing is conducted by the borrower independently of the FRA.10/4/2009Bahattin Buyuksahin, Celso Brunetti1515Forward Settlement in arrearsConsider what happens if the FRA in settled in March, on day 211, the loan repayment date. In that case, the payment to the borrower should be

If the borrowing rate is 1.5%

Since the rate is lower than FRA rate, the borrower pays the FRA counterparty.

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16FRA Settlement in arrearsIf the borrowing rate turns out to be 2%, the payment under FRA should be

What if the settlement occurs at the time of borrowing? If the FRA is settled in December, at the time of the money borrowed, payments will be less than when settled in arrears because the borrower has time to earn interest on the FRA settlement.

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17FRA settlement at the time of borrowingThe payment for a borrower is

If r=1.5%, then payment in December is

The future value of this is -$300,000. In order to make this payment, the borrower can borrow an extra $295,566.5, which results in an extra $300,000 loan payment in March.

If r=2%, then the payment is $196,078.43. The borrower can invest this amount, which gives $200,000 in March.

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18Forward Rate Agreement (continued)Consider a FAR where company X is agreeing to lend money to company

Value of a FRA:

Example:3-month LIBOR is 5%, 6-month LIBOR is 5.5%. Consider a FRA where you will receive 7% rate with quarterly compounding on a principal of $1 Million between 3 months and 6 months. The forward rate is 6.0452% (try to compute it). The value of the FRA is

1,000,000 *(0.07 0.060452)*(0.25)*exp(-0.055*0.5) = $2,32210/4/2009Bahattin Buyuksahin, Celso Brunetti19

19Interest Rate FuturesTreasury Bond FuturesEurodollar Futures10/4/2009Bahattin Buyuksahin, Celso Brunetti2020Day Count and Quotation ConventionThe day count defines in which interest accrues over time.The interest earned between the periods can be defined as

Actual/ actual (tbond) or 30/360 (municipal bond) or Actual/360 (money market instruments)For treasury bond actual/actual day count convention is used

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21Day Count and QuotationAssume bond principal $100, coupon payment dates are March 1 and September 1, the coupon rate 8%, what will be the interest earned between March 1 and July 3?

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22Price QuotationsDefine P is quoted price(or annualized discount yield), Y is the cash price, n is the remaining life of the bond, then

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23US Treasury BondsQuoted in dollars and thirty-seconds of a dollar for a bond with a face value of $100.A quote of 95.16 indicates that the quoted price for a bond with a face value of $100,000 is $95,500.Quoted price (clean price) is not the same as cash price (dirty price) paid by the purchaser of the bond.Cash Price=Quoted Price + Accrued interest since last coupon dateExample: Suppose it is September 30,2009 and the bond under consideration is an 5% coupon bond maturing in January 10, 2020, with a quoted price of 95.16. Because coupons paid semiannually, the most recent coupon date is July 10, 2009 and next one is January 10, 2010. The number of days between July 10, 2009 and September 30, 2009 is 82, whereas the number of day between July 10, 2009 and January 10, 2010 is 184. 10/4/2009Bahattin Buyuksahin, Celso Brunetti2424US Treasury BondOn a bond with face value of $100, the coupon payment is $2.5 on July 10 and January 10. The accrued interest on September 30, 2009 is the share of January 2010 coupon accruing to the bondholder on September 30, 2009. Because actual/actual in period is used for Tbond, this is $1.114. The cash price per $100 face value for the bond is therefore $95.5+1.114=96.61, which implies the cash price for $100,000 bond is $96,610.10/4/2009Bahattin Buyuksahin, Celso Brunetti2525Treasury Bond FuturesT-bond futures contract allows the party with the short position to choose to deliver any bond that has a maturity of more than 15 years and is not callable within 15 years. Cash price received by party with short position = Quoted futures price Conversion factor + Accrued interest10/4/2009Bahattin Buyuksahin, Celso Brunetti2626Conversion Factor The conversion factor for a bond is approximately equal to the value of the bond on the assumption that the yield curve is flat at 6% with semiannual compounding 10/4/2009Bahattin Buyuksahin, Celso Brunetti2727Conversion Factor10% coupon bond with 20 years and 2 months to maturity. For the purpose of computing the conversion factor we assume that the maturity is 20 years. The first coupon payment is assumed to be made after 6 months from now. Face value is $100. When the discount rate is 6% per annum with semiannual compounding (or 3% for 6 months), the value of the bond is:

Dividing by the face value, gives us the conversion factor: 1.462310/4/2009Bahattin Buyuksahin, Celso Brunetti28

28CBOTT-Bonds & T-NotesFactors that affect the futures price:Delivery can be made any time during the delivery monthAny of a range of eligible bonds can be deliveredThe wild card play10/4/2009Bahattin Buyuksahin, Celso Brunetti2929Cheapest to Deliver BondAt any given time during the delivery month, there are many bonds that can be delivered in the CBOT Tbond futures contract. The party with the short position can choose which of the available bonds is cheapest to deliver.If bond yield>6%, the delivery of low-coupon long maturity bonds.If bond yield